I would say one obstacle that stands in the way of "spying" on objects moving on the Earth's surface is that the gravitational wave energy emitted by accelerating objects on Earth would be "too small" for current detectors. Not to mention that there would be the issue of how to filter gravitational wave noise, and/or isolate frequencies. However, if it possible to build an amplifier or filter to resolve these issues, that remains to be seen - or maybe somebody else could chime in.

edit: typos, clarity

Funny choice of words. Screens were "on the radar" systems which prompted the development of one of the earliest forms of computer memory. https://en.wikipedia.org/wiki/Delay_line_memory

(I don't know if anyone envisioned shrinking those vats of mercury down to pocket size, however.)

Submarines already do this to detect other vessels AFAIK.